In this paper we give and discuss a counterexample related to the approximation of non-quadratic optimal control problems for linear distributed parameter systems. More precisely, we show that the assumption (1.7), which has been proved in a previous paper to be sufficient, is also necessary for the convergence (at least in the most general setting of the problem).
Ferretti, R. (1996). Convergence of semidiscrete approximations to optimal control problems in hilbert spaces: A counterexample. SYSTEMS & CONTROL LETTERS, 27(2), 125-128 [10.1016/0167-6911(95)00052-6].